TIDUF77 Design guide

TIDUF77 June 2024 – December 2024 MSPM0G1507

2.4.2.2.1.3 Rotor Position and Speed Estimation With PLL

With the arc tangent method, the accuracy of the position and velocity estimations are affected due to the existence of noise and harmonic components. To eliminate this issue, the PLL model can be used for velocity and position estimations in the sensorless control structure of the IPMSM. Section 2.4.2.2.1.2 illustrates the PLL structure used with SMO. The back-EMF estimations Equation 25 and Equation 26 can be used with a PLL model to estimate the motor angular velocity and position as shown in Figure 2-23.

Figure 2-23 Block Diagram of Phase-Locked Loop Position Tracker

Since Equation 36, Equation 37, and Equation 38, the position error can be defined as Equation 39:

Equation 36.

e_{α} = Ecos (θ_{e})

Equation 37.

e_{β} = Esin (θ_{e})

Equation 38.

E = ω_{e} λ_{pm}

Equation 39.

ε = {\hat{e}}_{β} \cos ({\hat{θ}}_{e}) - {\hat{e}}_{α} \sin ({\hat{θ}}_{e}) = Esin (θ_{e}) \cos ({\hat{θ}}_{e}) - Ecos (θ_{e}) \sin ({\hat{θ}}_{e}) = Esin (θ_{e} - {\hat{θ}}_{e})

where E is the magnitude of the EEMF, which is proportional to the motor speed ω_e.

When Equation 40, then Equation 39 can be simplified as Equation 41.

Equation 40.

(θ_{e} - {\hat{θ}}_{e}) < \frac{π}{2}

Equation 41.

ε = E (θ_{e} - {\hat{θ}}_{e})

Further, the position error after the normalization of the EEMF can be obtained (Equation 42):

Equation 42.

ε_{n} = θ_{e} - {\hat{θ}}_{e}

According to the analysis, the simplified block diagram of the quadrature phase-locked loop position tracker can be obtained as shown in Figure 2-24. The closed-loop transfer functions of the PLL can be expressed as Equation 43:

Equation 43.

\frac{{\hat{θ}}_{e}}{θ_{e}} = \frac{k_{p} s + k_{i}}{s^{2} + k_{p} s + k_{i}} = \frac{2 {ξω}_{n} s + ω_{n}^{2}}{s^{2} + 2 {ξω}_{n} s + ω_{n}^{2}}

where

k_p and k_i are the proportional and the integral gains of the standard PI regulator

The natural frequency ω_n and the damping ratio ξ are given in Equation 44:

Equation 44.

k_{p} = 2 ξ ω_{n}, k_{i} = ω_{n}^{2}

Figure 2-24 Simplified Block Diagram of Phase-Locked Loop Position Tracker